Abstract

The concepts of -remote neighborhood family, -cover, and -compactness are defined in -spaces. The characterizations of -compactness are systematically discussed. Some important properties of -compactness such as -closed heredity, arbitrarily multiplicative property, and preserving invariance under -continuous mappings are obtained. Finally, the Alexander -subbase lemma and the Tychonoff product theorem with respect to -compactness are given.

1. Introduction

Compactness is one of the most important notions in general topology, fuzzy topology, and -topology. Many research workers have presented various kinds of compactness [1–19] by means of introducing various operators, such as closure operator, -closure operator, -closure operator, -closure operator, -closure operator, -closure operator, and -closure operator; because the above operators are all order preserving. That is, they satisfy the following conditions: (i) if and , then ; (ii) for any , , where can take and of the above operators, is the family of all -sets defined on and with value in , is a fuzzy lattice, and is the greatest -set of . We introduced a kind of generalized fuzzy space called -space in [20] in order to unify various elementary concepts in -topological spaces. In the present paper, we will propose and study a generalized compactness which will be called -compactness in -spaces. The -compactness is a unified form of -compactness [16, 19], near -compactness [5], almost -compactness [6], -compactness [13], -compactness [1], -compactness [2], -compactness [9], -compactness [18], and so forth.

2. Preliminaries

Throughout this paper, denotes a fuzzy lattice, that is, a completely distributive lattice with order-reserving involution , 0 and 1 denote the least and greatest elements of , respectively, and denotes the set that consisting of all nonzero -irreducible elements of . Let be a nonempty crisp set, the set of all -fuzzy sets (briefly, -sets) on , and the set of all nonzero -irreducible elements (i.e., so-called molecules [17] or points for short) of . The least and the greatest elements of will be denoted by and , respectively. For any , is called the greatest minimal set of [12], and is said to be the standard minimal set of [17].

Definition 1 (Chen and Cheng [20]). Let be a nonempty crisp set.(i)An operator : is said to be an -operator if for all , and , ; for all , .(ii)An -set is called an -set if .(iii)Put , and call the pair () an -space.

Definition 2 (Chen and Cheng [20]). Let (, ) be an -space, , and . If there exists a such that and , then call an -remote neighborhood (briefly, -neighborhood) of . The collection of all -neighborhoods of is denoted by . If for each , then is said to be an -adherence point of and the union of all -adherence points of is called the -closure of and denoted by . If , then call an -closed set and call an -open set. If is an -closed set and , then is said to be an -closed remote neighborhood (briefly, -neighborhood) of and the collection of all -neighborhoods of is denoted by . Note that and are the family of all -closed sets and all -open sets in , respectively.

Definition 3 (Chen and Cheng [20]). Let () be an -space, , and and is an -open set in . We call the -interior of . Obviously, is -open if and only if .

Definition 4 (Huang and Chen [11]). Let () be an -space, let be a molecular net in , and let . If is eventually not in for each , then is said to be an -limit point of (or -converges to ). If is frequently not in for each , then is said to be an -cluster point of (or -accumulates to ). The union of all -limit points (-cluster points) of is written by -lim (-).

Definition 5 (Huang and Chen [11]). Let () be an -space, let be an ideal in , and let . If , then is called an -limit point of (or -converges to ). If for each and each , then is called an -cluster point of (or -accumulates to ). The union of all -limit points (-cluster points) of is denoted by -lim (-).

Definition 6 (Chen and Cheng [20]). Let () be an -space, , and , . Then,(i) is said to be an -base in () if for each , there exists a subfamily of such that ;(ii) is said to be an -subbase in () if the collection consisting of all intersections of any finite elements in is an -base in ().

Definition 7 (Chen and Cheng [20]). Assume (, ) to be an -space and an -valued Zadeh’s type function [17]. If for each , then call -continuous.

3. -Compact Set and Its Characteristics

In this section, we will introduce the concepts of -remote neighborhood family and -cover in an -space first, propose the notion of -compactness by making use of -remote neighborhood family next, and then discuss the characteristics of -compactness.

Definition 8. Suppose (, ) be an -space, , , and . If there exists a such that for each molecule in , then is called an -remote neighborhood family (briefly, -RF) of , in symbol . If there exists a nonzero -irreducible element with , then is said to be an -RF, in symbol .

Definition 9. Assume () be an -space, , , and . If there is a such that for each = , then is known as a -cover. If there exists a prime element such that is a -cover of , then is said to be a -cover of , where is the standard maximal set of [17].

Definition 10. Assume () be an -space and . If every -RF of has a finite subfamily such that is an -RF, where , then call an -compact set. If is an -compact set for any , then call an -compact set. Specially, when is -compact, we call () an -compact space, and if () is -compact for each , we say that () is an -compact space.

Obviously, when is the -closure operator on , the -compactness is just the -compactness in [19], and while takes the -closure operator (resp., -closure operator, -closure operator, -closure operator, -closure operator, and -closure operator) on , the -compactness is just the -compactness (resp., -compactness, near -compactness, -compactness, -compactness, and -compactness). Therefore, the -compactness is of the universal significance.

Example 11. Let () be an -space and . If the support of is a finite set, then is an -compact set.

Proof. Assume that and is an -RF of . For each we choose an -closed set with . Being , there is a such that . Since is an upper directed set, there is a with for each , and thus . Therefore has a finite subfamily which is an -RF of . By Definition 10, is an -compact set.

Now we give some characteristics of -compactness as follows.

Theorem 12. Let be an -space and . Then is an -compact set if and only if the following conditions hold:(1)for each , every -RF of has a finite subfamily with ;(2)for each , if is an -RF of , then is also an -RF of .

Proof. Necessity. Assume that is -compact and is an -RF of . According to Definition 10, has a finite subfamily with and so it certainly holds that . Thus is satisfied. If = is an -RF of , then has a finite with by the -compactness of . Obviously, , and hence is an (-RF of . Therefore holds.Sufficiency. Suppose that conditions and are satisfied, and is an -RF of (). By , there is a finite subfamily of such that is an -RF of . Let . Then is an -RF of . According to , is also an -RF of ; that is, there exists a with for each molecule . Since is finite, we can choose an -closed set with ; that is, . This shows that is an -RF of . Therefore is -compact.

Theorem 13. Let () be an -space and . Then is an -compact set if and only if for each , every -cover of has a finite subfamily such that is a -cover of .

Proof. Necessity. Suppose that is an -compact set and is any -cover of (). Put . Then , and there is an -closed set with for each ; that is, ; equivalently, . This implies that is a -RF of . Thus has a finite subfamily which is a -RF of ; that is, there exists such that for each we can take an -open set with . In other words, there are and with for each . This means that is a finite subfamily of and a -cover of .Sufficiency. Assume that every -cover of has a finite subfamily which is a -cover of (). If is an -RF of (), then is a -cover of where . Hence has a finite subfamily which is a -cover of by the hypothesis. Write . One can easily see that is a finite subfamily of and is an -RF of . Therefore is -compact.

Theorem 14. Let () be an -space and . Then is -compact if and only if for each and each having -finite intersection property for (i.e., for each finite subfamily of and each there exists a molecule with ), there exists a molecule with .

Proof. Necessity. Grant that is an -compact set, , and has -finite intersection property for (). If for each , then is an -RF of by the hypothesis of . Hence has a finite subfamily which is an -RF of ; that is, there is a satisfying for each ; in other words, . It contradicts the fact that has -finite intersection property for . Hence the necessity is proved.Sufficiency. Assume that the condition holds and that is an -RF of . If for any finite subfamily of , is not an -RF of , then for each there exists a molecule with ; that is, . This shows that has -finite intersection property for . By the assumption we have satisfying . It contradicts that is an -RF of . Therefore has a finite subfamily which is an -RF of , and hence is -compact.

Theorem 15. Let () be an -space and . Then is -compact if and only if for each , every -net in has an -cluster point in with height .

Proof. Necessity. Suppose that is an -compact set and that is an -net [16] in . If does not have any -cluster point in with height , then there exists a such that is eventually in for each ; that is, there is a with whenever . Write . Obviously, is -RF of . By the -compactness of , has a finite subfamily which is an -RF of ; that is, there is an with for some and each . Take . Then for each . Since is a directed set, there is an , such that and whenever , and so . This shows that for each , as long as . It contradicts the fact that is an -net. Therefore has at least an -cluster point in with height .Sufficiency. Assume that every -net in has at least an -cluster point with hight for each , is an -RF of , and is the set of all finite subfamilies of . If for each and each , is not an -RF of ; that is, for each , and hence there exists a molecule satisfying . In , we define the relation as follows: if and only if and , then is a directed set with the relation “”. Let . One can easily see that is an -net in . We assert that does not have any -cluster point in with hight . In fact, for each , we can choose an -closed set with by the definition of . Taking and , we have according to , and hence . This implies that is eventually in , and thus is not an -cluster point of . It is in contradiction with the hypothesis of sufficiency. Consequently, is -compact.

Definition 16. Let (, ) be an -space, let be an -filter in ; that is, for each and . If and for each and each , then is called an -cluster point of .

Theorem 17. Let () be an -space and . Then is -compact if and only if for each , every -filter containing as an element has an -cluster point in with hight .

Proof. Necessity. Grant that is an -compact set and that is an -filter containing as an element. Then for each and , and thus there exists a molecule with hight for each . Define and define a relation in as follows:
Evidently, is a directed set with the relation “”, and then is an -net in . By the -compactness of and Theorem 15, has an -cluster point in with hight , say . We assert that is also an -cluster point of . In reality, is frequently not in for each ; that is, for each there exist with and some satisfying . Hence we have by virtue of the fact that . This means that is an -cluster point of . Therefore the necessity is proved.Sufficiency. Suppose that every -filter containing as an element has an -cluster point in with hight for each and that is an -RF of . If for each , is not an -RF of , then there exists a molecule and for each . Put with . One can easily verify that is an -filter containing as an element, and hence has an -cluster point in with hight by the supposition, say . In accordance with Definition 16, we have for each and each , specially, . Since is an -RF of , there exists an -closed set with for each . Obviously, , so , and this is impossible. Hence there must be a which is an -RF of . This shows that is -compact.

Definition 18. Let be an ideal in . If for each , then is called an -ideal ().

Theorem 19. Let be an -space and . Then is -compact if and only if every -ideal whose is not in has an -cluster point in with hight for each .

Proof. Necessity. Assume that is an -compact set, is an -ideal whose is not in , and where and . Then is an -net in . Hence has an -cluster point in with hight by Theorem 15, say . Obviously, is also an -cluster point of . Consequently, the necessity is proved.Sufficiency. Grant that every -ideal whose is not in it has an -cluster point in with hight for each and is an -filter containing as an element. Let . Evidently, is an -ideal whose is not in . Now we will prove that has an -cluster point in with hight . Actually, by the hypothesis we know that has an -cluster point in with hight , say ; that is, ; equivalently, , for each and each . Therefore is an -cluster point of in line with Definition 16, and hence is an -compact set by Theorem 17. This implies that the sufficiency holds.

4. Some Important Properties of -Compactness

In this section, we still further deliberate the properties of -compactness in an -space.

Theorem 20. Let be an -space and . If is -compact and is -closed, then is -compact.

Proof. Assume that is an -net in . Then is also an -net in . Since is -compact, has an -cluster point in with hight , say . We assert that . Actually, since is an -net in and -accumulates , has an -subnet which -converges to and so . Hence , and thus is -compact in accordance with Theorem 15.

This theorem shows that the -compactness is hereditary with respect to -closed sets.

Theorem 21. Let and be both -compact sets in . Then is also an -compact set in .

Proof. Suppose that is an -RF of (). Then is an -RF of both and . Owing to the -compactness of , there are and with . Similarly, there exist and satisfying . Take and ; then , , and ; that is, is an -RF of . Consequently, is -compact.

This theorem indicates that the -compactness is finitely additive.

Theorem 22. Let , be an -space and let be an -compact set. Then there exists a crisp point such that .

Proof. Let ; then . If , then and hence holds for each . If , and is the set of all natural numbers, then we choose with and . Obviously, is an -net in , and has an -cluster point in by virtue of the -compactness of . Hence by . On the other hand, by the definition of . Therefore .

This theorem implies that an -compact set can reach the maximum at some point in as a function.

Theorem 23. Let and be an -space and an -space, respectively, and let be an -continuous -valued Zadeh’s type function. If is an -compact set in (), then is an -compact set in ().

Proof. Assume that is an -RF of and with . According to the definition of , there is a molecule such that and . Thus there is an -closed set with ; that is, . Since is ()-continuous, is -closed in (), and hence . This means that is an -RF of . Therefore has a finite subfamily such that is an -RF of . We assert that is an -RF of . In reality, there exists a with by virtue of the fact that is an -RF of . Since for each there exists a satisfying , and there exists a with , that is, . Hence by Lemma 3.1 in [19], and so is an -RF of . Consequently, is an -compact set in (, ).

This theorem means that the -compactness is topological variant under ()-continuous -valued Zadeh’s type functions.

Definition 24. Let (, ) be a crisp -space, and let be the set of all subsets of , that is, all crisp sets on and , where is a crisp -operator which satisfies the following conditions: for each and ; for each .(i)If , where denotes the set of all crisp -closed sets on and , then is said to be an -valued lower semicontinuous function on .(ii)Let (Ω) be the set of all -valued lower semicontinuous functions on , and call the pair (, ) the -space topologically generated by ().

Theorem 25. Let () be a crisp -space and let () be the -space topologically generated by (). Then is -compact if and only if is -compact for each .

Proof. Necessity. Provided that is an -compact set in () and is an -open cover of , let and , where is the characteristic function of . We assert that is a -cover of . In fact, for each , there is an -open set with ; that is, . Hence by virtue of the fact that is a prime element in with . Thus has a finite subfamily such that which is a -cover of in line with Theorem 13; that is, there is an such that with for some and each , and so . This implies that . Hence is an -compact set in (, ).Sufficiency. Grant that is an -compact set in (, ) for each and that is a -cover of where . Then there is an -open set with for each , and hence there exists a prime element satisfying . Put and ; then is an -open cover of according to and . Because of the -compactness of , has a finite subfamily which is an -open cover of ; that is, there exists an with ; in other words, for each . Take ; evidently, and . Hence is a -cover of , and thus is an -compact set in (, ) by Theorem 13.

This theorem indicates that the -compactness is a good extension in the sense of R. Lowen.

Theorem 26. Let (, ) be a stratified and . If is -compact, then is -closed.

Proof. We only prove that for each with by the definition of -operator. Actually, if , then there exists a molecular net in which -converges to in accordance with Theorem 2 in [11]. Write ; we assert that . In fact, if , then there is a with , and let . Since () is stratified, the constant -set on is -closed and , that is, . Obviously, is eventually in , and it contradicts the fact that -converges to . Hence ; that is, for each . For each and each we choose such that and , and define the relation “” in as follows:
Then is a directed set with the relation. Write ; then , where is defined as . Evidently is a subnet of and -converges to , and is an -net in . Being the -compactness of , has an -cluster point in with hight , say . Since (, ) is an space, -converges to and -accumulates to , by Theorem 2.7 in [11], and hence . This implies that ; that is, is an -closed set.

The following example shows that the stratified condition in Theorem 26 can not be omitted.

Example 27. Let be a single set, , and let be the fuzzy closure operator. Define , where is defined as , for . Obviously, () is both an -compact space and an -compact space. According to Example 11 we know that is an -compact set in (), but is not -closed.

The following theorems imply that the -compactness can strengthen -seperation properties.

Theorem 28. If (, ) is both and -compact -space, then () is an -regular space [11].

Proof. Let be an -closed pseudocrisp set and let be a molecule which is not in . By Definition 7.1 in [19], there is an such that implies . For each , there are and satisfying by virtue of and the separation of (). Put ; then is an -RF of . Since () is an -compact space, is an -compact set in accordance with Theorem 20, and thus has a finite subfamily which is an -RF of ; that is, there is an such that for each molecule we have with . Let ; then ; that is, for each . Since implies that , for each , and hence . Write ; then and
Consequently, (, ) is an -regular space.

Theorem 29. Let () be an -compact space. Then () is an -normal space [11].

Proof. Let both be -closed pseudocrisp sets in () with . Then there are , such that if and only if , and if and only if . According to the proof of Theorem 28, for each molecule , there is an -closed set satisfying for each , and there is a such that . One can easily see that is a -RF of . In line with Theorem 20 we know that is an -compact set, and so has a finite subfamily such that is a -RF of . Put ; ; then , and . Therefore (, ) is an -normal space.

5. The Tychonoff Product Theorem

In this section, we will first extend Alexandar’s subbase Lemma in general topology and give the Alexandar’s -subbase lemma and next prove that the Tychonoff product theorem holds in -spaces.

Theorem 30 (Alexandar -subbase lemma). Let (, ) be an -space, , and let be an -subbase [20] in . If for each -RF of where , there is a finite subfamily of with , then is -compact.

Proof. Suppose that is an arbitrary -RF of . We will prove that has a finite subfamily which is an -RF of . In fact, if for each , does not hold, then , for all does not hold, and is a partial-ordered set with respect to the upper bound and hence there exists a maximal element in by Zorn’s Lemma. We assert that satisfies the following conditions:(1);(2)if , then for each with ;(3)if and , then or . Actually, since and , condition holds. If , , , and is not in , then and . It contradicts the fact that is the maximal element in thus condition holds. Let . If and are both not in , then and are both not in by the maximality of , and thus there are such that and according to the definition of ; that is, there are with and . Since is upper directed, we can choose with . Now we prove . In reality, if does not have any -neighborhood of for each , then does not have any -neighborhood of and , respectively, and hence and . Particularly, and so . This shows that . Therefore is not in by virtue of the definition of and . So, condition holds.From and we have the following result: (4) If , and , then there is an satisfying .Consider now . If is an of , then there is a finite subfamily of which is an -RF of . Evidently, ; it is in contradiction with . Hence is not an -RF of ; that is, there is a molecule in meeting . We now verify that . In fact, if there is with , then by Definition 5 in [17] we can take a finite subfamily of satisfying , where is a finite set for each . Because of , we can choose with . Since , there is a such that by . Hence and ; it contradicts the fact that ; thus . However, this is in contradiction with again. This implies that has a finite subfamily with . Therefore is an -compact set in ().

Theorem 31. Let be a collection of -spaces and let () be the product space of them. If is an -compact set in for each , then the product of all -compact sets is an -compact set in ().

Proof. Assume that is an -RF of . By Theorem 30 we can grant that every -closed set in is of the form where and is a protection because is an -subbase in () [20]. Now we consider the following two cases.(i) If there exists a such that no molecule with hight is contained in , then by the -compactness of , there is an such that no molecule with hight is contained in . In reality, if there exists a molecule with hight in for each , say , then is an -net in by the directivity of . Since is -compact, has an -cluster point in with hight according to Theorem 15. It is in contradiction with the hypothesis of . Thus it can be seen that there exists an with for each . Hence for each , we have
and hence for each ; that is, no molecule with hight is contained in . This shows that for each , is an -RF of .(ii) Suppose that for each , contains a molecule with hight , say . Since , we can take such that , where . Now we prove that there must be with . In fact, if there is a crisp point such that for each , then we choose a crisp point in as follows: if , ; if is not in , . Taking any -closed set in , where and , we have
that is, , and hence by the arbitrariness of . On the other hand,
This implies that is a molecule in ; it contradicts the fact that is an -RF of . Consequently, there is with ; thus there is a finite subfamily of with for some . Put ; then . We assert that . Actually, for any molecule in with hight we have ; that is, is a molecule in , where . Hence there exists an -closed set meeting by virtue of the fact that is an -RF of ; thus ; that is, . This shows that is an -RF of . Therefore is an -compact set in ().

Theorem 32 (Tychonoff product theorem). Let (, ) be the product space of a collection of -spaces . Then (, ) is -compact if and only if for each , (, ) is -compact.

Proof. Necessity. Assume that (, ) is an -compact space. Since is an -continuous -valued Zadeh’s type function for each , is an -compact space by Theorem 23. Therefore the necessity holds.Sufficiency. It follows from Theorem 31.

The following example shows that the inverse theorem of Theorem 31 does not hold.

Example 33. Let be a countably infinite set, for each , , and let be a fuzzy closure operator. Then is a discrete -space for each . Define () as follows:if , ; if , . Suppose that () is the product space of and . Now we prove that is an -compact set in (, ), but is not an -compact set in for each . In reality, for each we put , where is a crisp point in ; then from the definitions of and fuzzy product set